Volume 2005, Issue 585

Let ϕ ( z ) be a polynomial of degree at least 2 with coefficients in a number field K . Iterating ϕ gives rise to a dynamical system and a corresponding canonical height function ĥ ϕ , as defined by Call and Silverman. We prove a simple product formula relating the transfinite diameters of the filled Julia sets of ϕ over various completions of K , and we apply this formula to give a generalization of Bilu’s equidistribution theorem for sequences of points whose canonical heights tend to zero.

We study the varieties of reductions associated to the four Severi varieties, the first example of which is the Fano threefold of index 2 and degree 5 studied by Mukai and others. We prove that they are smooth but very special linear sections of Grassmann varieties, and rational Fano manifolds of dimension 3 a and index a  + 1, for a  = 1, 2, 4, 8. We study their maximal linear spaces and prove that through the general point pass exactly three of them, a result we relate to Cartan’s triality principle. We also prove that they are compactifications of affine spaces.

We study the structure of the Eisenstein component of Hida’s ordinary p -adic Hecke algebra attached to modular forms , in connection with the companion forms in the space of modular forms (mod p ). We show that such an algebra is a Gorenstein ring if certain space of modular forms (mod p ) having companions is one-dimensional; and also give a numerical criterion for this one-dimensionality. This in part overlaps with a work of Skinner and Wiles; but our method, based on a work of Ulmer, is totally different. We then consider consequences of the above mentioned Gorenstein property. We especially discuss the connection with the Iwasawa theory.

For A | F a central simple algebra over a p -adic local field the group of units A × ≅ GL m ( D d ) is a general linear group over a central division algebra D d | F of index d . The product n  = dm being fixed, the Abstract Matching Theorem (AMT) implies the existence of bijective maps ℐℒ between the sets of discrete series representations of the groups A × such that a character relation is preserved. In this paper we construct maximal level zero extended type components for every level zero discrete series representation of A × . Its maximal level zero extended type determines the discrete series representation uniquely (without any twist ambiguities as for the usual types) and, conversely, every level zero discrete series representation ∏ contains a maximal level zero extended type component ## add figure 'crll.2005.585.173_01.gif' ## which is unique up to conjugacy. In order to determine how ℐℒ matches the extended types we find certain regular elliptic elements where the characters of ## add figure 'crll.2005.585.173_01.gif' ## and ∏ are the same and we compute the character values at these elements by using a version of Shintani descent which we develop in Appendix B. Surprisingly, we find that AMT also implies explicit Shintani descent for irreducible characters of finite general linear groups which have cuspidal descent.